Question

For the matrix A satisfying the equation given below, the eigen values are
$\left[A\right]\left[\begin{array}{ccc}1& 2& 3\\ 7& 8& 9\\ 4& 5& 6\end{array}\right]=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$

• A. (1, -j, j)
• B. (1, 1, 0)
• C. (1, 1, -1)
• D. (1, 0, 0)

Answer 1

The correct option is C (1, 1, -1)

The given matrix equation can be written as
AB = C ....(i)
Notice that, the matrix 'C' can be obtained from 'B' by interchanging $R_2$ and $R_3$

Let $I=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

then $E_{23}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right]$ using $R_2\leftrightarrow R_3$

Now $E_{23}B=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right]\left[\begin{array}{ccc}1& 2& 3\\ 7& 8& 9\\ 4& 5& 6\end{array}\right]$
$=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$

i.e. $E_{23}B=C$ ...(ii)

Comparing (i) and (ii),

$A=E_{23}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right]$

Now, Characteristic equation of A is $|A-\lambda I|=0$
$\Rightarrow \left|\begin{array}{ccc}1-\lambda & 0& 0\\ 0& -\lambda & 1\\ 0& 1& -\lambda \end{array}\right|=0$
$\Rightarrow$ $(1-\lambda )({\lambda }^2-1)=0$
$\Rightarrow$ $\lambda =1,1,-1$